1. Department of Materials Science and Engineering Magnetization, Decay, and Influence on Error Fields in HTS accelerator magnets -- We thank Advanced Conductor Technologies and University of Colorado, Department of Physics, Boulder, CO, USA D. van der Laan Center for Superconducting & Magnetic Materials (CSMM), The Ohio State University, Columbus, OH, USA M. D. Sumption, C. Myers, M. Majoros, E. W. Collings W. Goldacker This work was funded by the U.S. Dept. of Energy, Division of High Energy Physics, under Grant No. under Grants No. DE- SC0010312 & DE-SC0011721 (OSU) (University Program) We thank OST for Nb 3 Sn and Bi:2212

2. Department of Materials Science and Engineering Motivation – error fields and drift While near-term (luminosity) LHC upgrades will require only NbTi and/or Nb 3 Sn, the subsequent energy upgrades of the LHC are expected to require HTS/LTS hybrids In addition, a high energy future circular collider may require YBCO or Bi:2212 (and Nb 3 Sn) The fact that strand and cable magnetization lead to field errors is well known, tools for predicting likely error components in a more global way is needed. This includes YBCO, Bi:2212, and ongoing Nb 3 Sn assessment Below we will start with NbTi dipole magnet characteristics, and explore the relative magnetic properties of Nb 3 Sn, YBCO, and Bi:2212 To key collider issues will be, what is the size of b3 (the sextupole error field), and what might the use of HTS do to drift in b3

3. Department of Materials Science and Engineering Field Errors at collision and injection In accelerator operation, there is (1) a low field injection phase, where the dipole magnets operate for some 20 minutes or more at a nominal 1 T injection “porch” (2) An energy ramp, coast (beam collision) (3) An energy dump, where the magnets are then cycled back to near zero, followed by a rise to the beam injection field to repeat the cycle Any strand magnetization leads to deviations from the pure dipole field which tend to defocus the beam. Such errors are described in terms of the high order multipoles of the field, a good measure is the sextupole component, b3 The above process leads to a partial compensation of the error fields at injection, a (hopefully small) negative value usually dominated by the sextupole component, b3. “Superconducting Accelerator Magnets” by P Ferracin, ETodesco, S O. Prestemon and H Felice, January 2012

4. Department of Materials Science and Engineering Magnetization Error Criteria The magnet designers criteria for magnet quality is terms of the multipole error fields, a good error to key in on for dipoles in the sextupole component, b 3. b 3 should be about “1 unit” or less 1 unit is 0.01% of the main field The correlation between strand magnetization and error fields is magnet design dependent cos theta dipoles and block magnets of different designs magnet cross section will have different magnetization to error field correlations Amemiya, IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 20, NO. 3, JUNE 2010 Different “stackings” of HTS can change error contributions

5. Department of Materials Science and Engineering NbTi Mag as a compare to 0.5 T, 15 mT => 12000 A/m From d f and J c, at 0.54, the values M coup,LHC = 2.64 kA/m and M sh,inj,LHC = 10.3 kA/m, may be regarded as bench-marks against which the corresponding properties of Nb 3 Sn cables can be compared. IEEE Trans. 9 1999 “Superconducting Accelerator Magnets” by P Ferracin, ETodesco, S O. Prestemon and H Felice, January 2012 b 3 is about 3, and M about 10 kA/m – but this will depend on magnet design. Is there a way to separate or partially separate magnet design and strand influence?

6. Department of Materials Science and Engineering Simple Model of b 3 ratio estimation---Partial separation of strand and magnet variables Any given dipole magnet will have a set of multipoles that depend on conductor magnetization in a design-specific way. This is in particular true for b 3, thus Here M sh,i is the magnetization of the i th part of the winding, located at position x,y,z, V i is the volume of a small region surrounding this point, b 3,i is the contribution of this region to the total b 3, and  i is the coupling between these two. Then, Where  and M sh are now continuous that depend on position within the winding. This can then be written where Here and below, the error field b 3 is given in units, and ,  are for a specific injection field For LHC  = 3 x 10 -4 units*m/A

7. Department of Materials Science and Engineering Angular Anisotropy, field dependence, history M sh (0) is the magnetization at injection, a field alignment for maximum magnetization, and standard field history. Then M sh (B)=M sh (0)(J c (B)/J c (0))  (x,y,z)  (x,y,z) = M sh (0)  (B{x,y,z},  {x,y,z})  (  {x,y,z})  (x,y,z,B p {  }) here  =J c (B)/J c (0),  represents the orientation of the local segment of conductor with respect to a reference orientation  (0<  <1) characterizes the fraction of the magnetization at a given angle relative to the position of maximum magnetization (purely geometric), and  (-1<  <1) represents the influence of field history on M sh, being = 1 for field sweep protocols where B is swept to B 0 -2B p or greater before being brought up to B 0, then

8. Department of Materials Science and Engineering Weighted  We can also define where  is a constant (with units –units*m/A) connecting b 3 to a reference magnetization is a related constant (units- units* A -1 m -2 ) connecting b 3 to a reference moment We will not attempt to calculate  or in detail, nevertheless, the former will have some utility for estimating b 3 from strand magnetizations, the latter is computational useful, as we will see below.

9. Department of Materials Science and Engineering Equal Volume Replacement Let us consider LHC-type dipoles in which the NbTi cable is replaced by equal volumes of Nb 3 Sn, Bi:2212, and YBCO-wound cables, then Where again Let the replacement conductor is isotropic (true for NbTi, Nb 3 Sn, and Bi:2212 conductors, CORC, and TSTC. In this case the  dependencies drop out, and  =1. Let us, for the sake of a direct comparison, set  ’(x,y,z) =  (x,y,z), i.e., we assume the same magnet ramp sequence will be used for the two magnets.

10. Department of Materials Science and Engineering Equal Volume Replacement II The integral terms are then essentially the field dependence of J c -weighted integrals of . If we choose a linear field dependence for  (referenced to B 0 ), then  =1+b[(B 0 -B)/B 0 ], then Where the second term is just the different in the linear coefficient in the Jc dependence of the two conductors multiplied by a factor which weights their contributions over the magnet. This term will be small (< 1) and positive; if it is << 1, then we find the simple approximation that b 3’ /b 3  M sh ’(0)/M sh (0). Then

11. Department of Materials Science and Engineering Magnetization and field error for “Equal volume replacement”

12. Department of Materials Science and Engineering Target Field Replacement Under the equal volume replacement, a lower Jc strand lead to a lower magnetization – but also a lower bore field Now, however, let us imagine a scenario where we want to replace a magnet made from NbTi strand with one made from a different stand type – but the target field is equal to or higher than the NbTi magnet, then

13. Department of Materials Science and Engineering Target Field Result An isotropic Jc with a linear B dependence leads to An isotropic Jc with an exponential B dependence leads to If we make no simplifying assumptions Or

14. Department of Materials Science and Engineering Result for field target However, V mag = L mag A mag where L mag is the length of the magnet and A mag is the cross-sectional area of the magnet, giving us, upon substitution (assuming that magnet length remains constant), The magnet load line is given by B bore = CI mag and B bore ′ = C’I mag ′, where C and C’ are magnet constants, and I mag is the current in the whole of the magnet cross section, such that I mag = N*I strand, and N is the number of strands in the magnet cross section. The C is (1/N) times the magnet constant as defined with respect to I strand. Then I mag = A mag *J mag. Then Which leads to

15. Department of Materials Science and Engineering Magnetization and field error “Target Field Replacement” With C’/C = 1 and  =1

16. Department of Materials Science and Engineering Some Clarifications HTSC magnets are presumably to be hybrids, using, say Nb 3 Sn to reach 15 T, and HTS to get to 20T. Also YBCO will be used in some mixed field orientation The final effective “average” contribution could be a 75%/25% mix – pushing YBCO/Nb 3 Sn hybrids to significantly lower mag – However, the HTS will be presumably closer to the beam. One final Note: While striating the YBCO will not influence Roebel cable Magnetization, it will reduce CORC and TSTC cable magnetization. From the above considerations, it seems that the striation target should be 100 filaments in a 4 mm wide strand

17. Department of Materials Science and Engineering Drift on the injection Porch Just as important as the absolute value of b 3 is any change with time during the injection porch It is possible to compensate for error fields with corrector coils, but the presence of drift makes this much more difficult At right is shown the drift of the error fields as a function of time from zero to 1000 seconds for LHC magnets, followed by a snap-back once the energy ramp begins The underlying mechanism for drift in NbTi magnets is the decay of coupling currents, (especially inhomogeneous and long length scale coupling currents) and their influence on the strand magnetization Need to keep both b 3 and its drift below 1 unit For NbTi and Nb 3 Sn based magnets, this is possible

18. Department of Materials Science and Engineering Important to control drift So, right now we are at several units of drift Drift in LTS is due to influence of long range coupling current decay on strand magnetization But HTS materials famously exhibit Giant Flux Creep (Y. Yeshurun and A. P. Malozemoff) But, Creep goes like kT, so its not a problem at Low Temperatures, right? No – at right is data of PRB paper at 4 K Even though creep reduced, still significant Especially for precision field

19. Department of Materials Science and Engineering Magnetization Creep and b3 If we use the Bi:2212 projection of b3 = 35 units, then a 20% change is 7 units If we use the YBCO projection of b3 = 500-1000 units, then a 15% change is 75-150 units. Downrated for insert, 20-40 units

20. Department of Materials Science and Engineering Summary and Conclusions Projected Accelerator injection Magnetization contributions for YBCO, Bi:2212, and Nb 3 Sn have been compared against NbTi directly D eff is not the whole story A simple method for correcting for conductor J c (at collision and injection), and the magnet target field leads to a more useful metric, assuming magnet design remains “constant”. Bi:2212 strands will cause lower magnetization errors than expected on a deff basis alone, because of the flatness of J c vs B. YBCO contributions are highly dependent on the amount of field perpendicular to the tape, but a striation target was suggested : 100 filaments/4 mm width Potential contributions to drift on the injection porch from creep (as opposed to the standard LTS mechanism) in Bi:2212 and YBCO were considered 10-20% changes in magnetization over a 20 minute time window were seen at 4K, for an 18 stack Bi:2212 with OK d eff and moderate magnetization, this leads to 1 unit or so – maybe OK For YBCO, tending to have higher Magnetization, 30-40 units was seen, would be diluted by hybrid implementation or parallel field orientation The above striation target, implemented in CORC cable, would make drift also OK for YBCO

21. Department of Materials Science and Engineering A. Xu, J. J. Jaroszynski, F. Kametani, Z. Chen, D. C. Larbalestier, Y. L. Viouchkov, Y. Chen, Y. Xie and V. Selvamanickam SUST, vol. 23, p. 014003, 2010 J c (B) curves of YBCO CCs Dropping from 77 K SF to 4.2 K SF increases J c by x 10. Increasing from SF to 20 T decreases  by about 10 Increasing from SF to 1 T, about 2 times less So, we expect M inj  500 kA/m M 20 T  100 kA/m This is expected for any field orientation, since cable is cylindrically symmetric